Question

Hi,
I'm trying to learn factoring but a couple of things don't make sense to me and I'm hoping you could explain them.

If I have P(x) = 2(x-3)(x-4), how do I automatically know that the zeros are x = 3 and x =4?

If inside the parentheses it was (x+3)(x+4), would the zeros then be x = -3 and x = -4?

Also, how do I determine the x intercepts off of this information? The answers say the x intercepts are (3,0) and (4, 0), but how do I determine them?

Thank you.

Answer 1

When you have a function such as
P(x) = 2(x-3)(x-4)
you could also write it as y = 2(x-3)(x-4)

So when they speak of the zeros of the function, they are really describing the x's that make y = 0
that is:
2(x-3)(x-4) = 0

Now think about the answer of 0 after a multiplication.
How can we possible get zero after a multiplication??
As long as we are multiplying non-zero numbers we can NEVER get 0 as an answer.
The only way is if one of our multipliers is 0
e.g. (4)(6)(8)(0)(12) = 0

but in 2(x-3)(x-4) = 0, we don't know which factor produced the zero answer.
It certainly couldn't be the 2, since 2 ≠ 0
It could have been the x-3
that is, x-3 = 0 ----> x = 3
OR
it could have been the x+4
then x+4 = 0 ---> x = -4

The zeros of the function are really the x-intercepts, since any point on the x axis has its value of y = 0

Hope this helps a bit